Determinant lines, von Neumann algebras and $L^2$ torsion

TL;DR

This paper constructs determinant lines for Hilbertian modules over von Neumann algebras and explores their relation to $L^2$ torsion invariants, connecting classical torsion with modern $L^2$ theories.

Contribution

It introduces a new construction of determinant lines for Hilbertian modules over von Neumann algebras and relates $L^2$ torsion invariants to classical torsion and volume forms.

Findings

Determinant lines can be viewed as volume forms on Hilbertian modules.

$L^2$ torsion invariants generalize classical Reidemeister and Ray-Singer torsions.

A reformulation of Burghelea et al.'s theorem as an equality of volume forms.

Abstract

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced $L^2$ homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the $L^2$ homology vanishes, the determinant line can be canonically identified with $\R$, and our $L^2$ torsion invariants specialize to the $L^2$ torsion invariants previously constructed by A.Carey, V.Mathai and J.Lott. We also show that a recent theorem of Burghelea et al. can be reformulated as stating equality between two volume forms (the combinatorial and the analytic) on the reduced $L^2$ cohomology.